The effects of a higher vorticity moment on a variational problem
for barotropic vorticity on a rotating sphere is examined rigorously
in the framework of the Direct Method. This variational model
differs from previous work on the Barotropic Vorticity Equation
(BVE) in relaxing the angular momentum constraint, which then allows
us to state and prove theorems that give necessary and sufficient
conditions for the existence and stability of constrained energy
extremals in the form of super and sub-rotating solid-body steady
flows. Relaxation of angular momentum is a necessary step in the
modeling of the important tilt instability where the rotational axis
of the barotropic atmosphere tilts away from the fixed north-south
axis of planetary spin. These conditions on a minimal set of
parameters consisting of the planetary spin, relative enstrophy and
the fourth vorticity moment, extend the results of
previous work and clarify the role of the higher vorticity moments in models of geophysical flows.